Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $z = \dfrac{-8k + 32}{k + 7} \div \dfrac{k - 4}{k^2 + 10k + 21} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{-8k + 32}{k + 7} \times \dfrac{k^2 + 10k + 21}{k - 4} $ First factor the quadratic. $z = \dfrac{-8k + 32}{k + 7} \times \dfrac{(k + 7)(k + 3)}{k - 4} $ Then factor out any other terms. $z = \dfrac{-8(k - 4)}{k + 7} \times \dfrac{(k + 7)(k + 3)}{k - 4} $ Then multiply the two numerators and multiply the two denominators. $z = \dfrac{ -8(k - 4) \times (k + 7)(k + 3) } { (k + 7) \times (k - 4) } $ $z = \dfrac{ -8(k - 4)(k + 7)(k + 3)}{ (k + 7)(k - 4)} $ Notice that $(k - 4)$ and $(k + 7)$ appear in both the numerator and denominator so we can cancel them. $z = \dfrac{ -8\cancel{(k - 4)}(k + 7)(k + 3)}{ \cancel{(k + 7)}(k - 4)} $ We are dividing by $k + 7$ , so $k + 7 \neq 0$ Therefore, $k \neq -7$ $z = \dfrac{ -8\cancel{(k - 4)}\cancel{(k + 7)}(k + 3)}{ \cancel{(k + 7)}\cancel{(k - 4)}} $ We are dividing by $k - 4$ , so $k - 4 \neq 0$ Therefore, $k \neq 4$ $z = -8(k + 3) ; \space k \neq -7 ; \space k \neq 4 $